If κ is an infinite cardinal number, then cf ( κ ) is the least cardinal such that there is an unbounded function from it to κ ; and cf ( κ ) = the cardinality of the smallest collection of sets of strictly smaller cardinals such that their sum is κ ; more precisely

However, if an inaccessible cardinal κ is assumed, then the sets of smaller rank form a model of ZF ( a Grothendieck universe ), and its subsets can be thought of as " classes ".

More generally, if κ is any infinite cardinal, then a product of at most 2 < sup > κ </ sup > spaces with dense subsets of size at most κ has itself a dense subset of size at most κ ( Hewitt – Marczewski – Pondiczery theorem ).

To achieve this, let T be the initial theory and let κ be any cardinal number.

The generalized Suslin hypothesis says that for every infinite regular cardinal κ every tree of height κ either has a branch of length κ or an antichain of cardinality κ.

However, κ does not need to be inaccessible, or even a cardinal number, in order for V < sub > κ </ sub > to be a standard model of ZF ( see below ).

In the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal μ, there is an inaccessible cardinal κ which is strictly larger, μ < κ.

It would challenge sharply not the cult of the motor car itself but some of its ancillary beliefs and practices -- for instance, the doctrine that the fulfillment of life consists in proceeding from hither to yon, not for any advantage to be gained by arrival but merely to avoid the cardinal sin of stasis, or, as it is generally termed, staying put.

I wrote a few years ago that one of the cardinal rules of writing is that the reader should be able to get some idea of what the story is about.

The only cardinal sin which may be committed in warming a wine is to force it by putting it next to the stove or in front of an open fire.

He has a pleasant sense of humor and is modest enough to admit mistakes and even `` a cardinal error ''.

It is also consistent with ZF + DC that every set of reals is Lebesgue measurable ; however, this consistency result, due to Robert M. Solovay, cannot be proved in ZFC itself, but requires a mild large cardinal assumption ( the existence of an inaccessible cardinal ).

ZF + DC + AD is consistent provided that a sufficiently strong large cardinal axiom is consistent ( the existence of infinitely many Woodin cardinals ).

One of the highlights of the facade is a tower topped with a cross of four arms oriented to the cardinal directions.

Related to the argument from morality is the argument from conscience, associated with eighteenth-century bishop Joseph Butler and nineteenth-century cardinal John Henry Newman.

This is a building with circular tower and doors facing the cardinal directions.

; Cardinal: In Roman Catholicism, a cardinal is a member of the clergy appointed by the pope to serve in the College of Cardinals, the body empowered to elect the pope ; however, on turning 80 a cardinal loses this right of election.

Under modern canon law, a man who is appointed a cardinal must accept ordination as a bishop, unless he already is one, or seek special permission from the pope to decline such ordination.

The Roman Breviary has undergone several revisions: The most remarkable of these is that by Francis Quignonez, cardinal of Santa Croce in Gerusalemme ( 1536 ), which, though not accepted by Rome ( it was approved by Clement VII and Paul III, and permitted as a substitute for the unrevised Breviary, until Pius V in 1568 excluded it as too short and too modern, and issued a reformed edition ( Breviarium Pianum, Pian Breviary ) of the old Breviary ), formed the model for the still more thorough reform made in 1549 by the Church of England, whose daily morning and evening services are but a condensation and simplification of the Breviary offices.

The walls defining the enclosures of Khmer temples are frequently lined by galleries, while passage through the walls is by way of gopuras located at the cardinal points.

" Venerable / Heroic in Virtue " When enough information has been gathered, the congregation will recommend to the pope that he make a proclamation of the Servant of God's heroic virtue ( that is, that the servant exhibited the theological virtues of faith, hope and charity, and the cardinal virtues of prudence, justice, fortitude and temperance, to a heroic degree ).

* The cardinal is also a fairy chess piece, also known as the archbishop

Two sets have the same cardinal number if and only if there is a bijection between them.

A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers and the set of natural numbers do not have the same cardinal number.

There is a transfinite sequence of cardinal numbers:

If α is a limit ordinal, an α-inaccessible is a fixed point of every ψ < sub > β </ sub > for β < α ( the value ψ < sub > α </ sub >( λ ) is the λ < sup > th </ sup > such cardinal ).

( Note that by designating cardinal directions for 1 ,-1,, and, complex numbers such as are considered constructible.

Confucianism holds that one should give up one's life, if necessary, either passively or actively, for the sake of upholding the cardinal moral values of ren and yi.

Confucianism holds that one should give up one's life, if necessary, either passively or actively, for the sake of upholding the cardinal moral values of ren and yi.

In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality ( size ) of sets.

Several influential stylebooks, both secular and religious, however, indicate that the correct form for referring to a cardinal in English is as " Cardinal < Name > < Surname >.

In 1965 Pope Paul VI decreed in his motu proprio Ad Purpuratorum Patrum that patriarchs of the Eastern Catholic Churches who were named cardinals would also be part of the episcopal order, ranked after the six cardinal bishops of the suburbicarian sees ( who had been relieved of direct responsibilities for those sees by Pope John XXIII three years earlier ).

Cardinal deacons have long enjoyed the right to " opt for the order of cardinal priests " ( optazione ) after they have been cardinal deacons for 10 years.

They may on such elevation take a vacant " title " ( a church allotted to a cardinal priest as the Roman church with which he is associated ) or their diaconal church may be temporarily elevated to a cardinal priest's " title " for that occasion.

Until 1917 it was possible for someone who was not a priest, but only in minor orders, to become a cardinal ( see " lay cardinals ", below ), but they were enrolled only in the order of cardinal deacons.

For example, in the 16th century, Reginald Pole was a cardinal for 18 years before he was ordained a priest.

When in choir dress, a Latin-rite cardinal wears scarlet garments — the blood-like red symbolizes a cardinal's willingness to die for his faith.

The biretta of a cardinal is distinctive not merely for its scarlet color, but also for the fact that it does not have a pompon or tassel on the top as do the birettas of other prelates.

Later modifications to his theory allowed for an additional set of eight " secondary Cardinal Vowels " with reverse lip shapes, permitting the representation of eight secondary cardinal vowels ( front rounded and back unrounded ).

For example, an equivalence relation with exactly two infinite equivalence classes is an easy example of a theory which is ω-categorical, but not categorical for any larger cardinal number.

For any ordinal α, a cardinal κ is α-hyper-inaccessible if and only if κ is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than κ is unbounded in κ.

Firstly, a cardinal κ is inaccessible if and only if κ has the following reflection property: for all subsets U ⊂ V < sub > κ </ sub >, there exists α < κ such that is an elementary substructure of.

This is close to being best possible, because the existence of 0 < sup >#</ sup > implies that in the constructible universe there is an α-Erdős cardinal for all countable α, so such cardinals cannot be used to prove the existence of 0 < sup >#</ sup >.

A cardinal number κ is called Π-indescribable if for every Π < sub > m </ sub > proposition φ, and set A ⊆ V < sub > κ </ sub > with ( V < sub > κ + n </ sub >, ∈, A ) ⊧ φ there exists an α < κ with ( V < sub > α + n </ sub >, ∈, A ∩ V < sub > α </ sub >) ⊧ φ.

If α is an ordinal, the cardinal number κ is called α-indescribable if for every formula φ and every subset U of V < sub > κ </ sub >

Continuing in this manner, it is possible to define a cardinal number for every ordinal number α, as described below.

" A cardinal κ is extendible if and only if for all α > κ there exists β and an elementary embedding from V ( α ) into V ( β ) with critical point κ.

The Erdős cardinal κ ( α ) is defined to be the least cardinal such that for every function

A cardinal κ is called subtle if for every closed and unbounded C ⊂ κ and for every sequence A of length κ for which element number δ ( for an arbitrary δ ), A < sub > δ </ sub > ⊂ δ there are α, β, belonging to C, with α < β, such that A < sub > α </ sub >= A < sub > β </ sub >∩ α.

A cardinal κ is called ethereal if for every closed and unbounded C ⊂ κ and for every sequence A of length κ for which element number δ ( for an arbitrary δ ), A < sub > δ </ sub > ⊂ δ and A < sub > δ </ sub > has the same cardinal as δ, there are α, β, belonging to C, with α < β, such that card ( α )= card ( A < sub > β </ sub >∩ A < sub > α </ sub >).